after the black hole emits radiation, the information in it is lost, how does it breaks the law of quantum theory? and what type of information it is that's inside them?
If I understand the problem correctly, it goes something like this... Suppose that an apple falls into a black hole. An apple contains information about it's "appleness". It has a certain structure. It is made of certain elements. Etc. Once the apple is inside the black hole, all that information is "lost" to the outside universe. It can't be extracted again from the hole. But wait! What about Hawking Radiation? Hawking predicted that, given a long enough time, black holes must gradually "evaporate", converting their mass into emitted particles (light, electrons, other particles). The problem is that the emitted radiation doesn't seem to contain all of the information from the apple any more. So, it looks like information has been lost from the universe, rather than conserved. Apparently, this is considered problematic, for reasons I don't know. The good news is that Hawking claimed to have solved the problem, finding that information is preserved after all. I'm not aware of whether the relevant research community accepts Hawking's proof that information is preserved. Provisionally, I'm going to assume they do, or else I probably would have heard more about the "unsolved" problem. Does this help?
Wiki has a primer on this: "... this violates a core precept of both classical and quantum physics—that, in principle, the state of a system at one point in time should determine its value at any other time. Specifically, in quantum mechanics the state of the system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future." I think the long and the short of it is that: 1. In principle - a "sufficiently accurate" list of the properties of a system can be used to recreate the evolution of the system either forward in time or backward in time. (I guess that's a deterministic universe). 2. But the radiation that comes out of a black hole is bereft of the properties required to roll it backward in an attempt to recreate the previous system.
